In this paper, we consider the system 8>>< >>: Δp(x)u = a(x)jujr1(x)−2u b(x)juj (x)−2u x 2 Ω Δq(x)v = c(x)jvjr2(x)−2v d(x)jvj (x)−2v x 2 Ω u = v = 0 x 2 @Ω where Ω is a bounded domain in RN with smooth boundary, ; > 0, p, q, r1, r2, and are continuous functions on ¯Ω satisfying appropriate conditions. We prove that for any > 0, there exists ∗ sufficiently small, and ∗ large enough such that for any 2 (0; ∗) [ ( ∗ ;1), the above system has a nontrivial weak solution. The proof relies on some variational arguments based on the Ekeland’s variational principle and some adequate variational methods.
